Describing Probability

Outcomes, Sample Spaces and Probability Models

Probability Outcomes

Define an outcome as the result of a random event. For example: an outcome of a coin flip might be HEADS:

1. What are the possible “suits” in a standard deck of playing cards?

2. How many playing cards are in each suit?

Describing Probability Outcomes

Select a card at random from a well-shuffled, standard 52-card deck of playing cards. What is the probability of selecting a card that belongs to a given suit (HEARTS, for example)? Before you could answer that question, you would probably have 2 questions of your own:

1. What are the possible “suits” in a standard deck of playing cards?

2. How many playing cards are in each suit?

Standard Deck of Playing Cards

52 cards, 4 suits, 13 cards per suit Ace-10, Jack, Queen, King (face cards)

Probability Sample SpaceLet’s begin with the first question:

What are the possible “suits” in a standard deck of playing cards?

The answer is what is called a probability sample space, which is a description, or list of all possible outcomes for a given random event.

If we call this list or set of outcomes, S, we can write S as

S = {Clubs, Spades, Hearts, Diamonds}

Probability ModelNow for the second question:

How many playing cards are in each suit?

The answer here leads us to what is called a probability model, which includes the sample space AND an assignment of probabilities to each outcome in that sample space.

Since there are 13 cards in each of the four suits, each suit has equal probability of being chosen. That probability is 13/52 = ¼ = 0.25.

Probability ModelsWe generally express a probability model in the form of a table, as follows:

Outcome Clubs Spades Hearts Diamonds

Probability ¼ ¼ ¼ ¼

Outcome Clubs Spades Hearts Diamonds

Probability .25 .25 .25 .25

Outcome Clubs Spades Hearts Diamonds

Probability 13/52 13/52 13/52 13/52

OR in reduced fraction form

OR in decimal form

More Examples - coins

Sample Space for the toss of a coin:

S = {Heads, Tails}

Outcome Head Tails

Probability ½ = 0.5 ½ = 0.5

Probability Model for the toss of a coin:

More Examples - dice

Sample Space for the roll of a 6-sided die:

S = {1, 2, 3, 4, 5, 6,}

Outcome 1 2 3 4 5 6

Probability 1/6 1/6 1/6 1/6 1/6 1/6

Probability Model for the roll of a die:

More Examples – diceA four-sided die or tetrahedral die is created to make all 4 sides equally likely to land face down. Each face has 3 numbers and the number facing up represents the result of that die roll. For the purple die at top right, the result is a 1. At bottom right, the green die is a 3.

A probability model for this die is:

Outcome 1 2 3 4

Probability .25 .25 .25 .25

More Examples – 2 diceNow consider a probability model for rolling two four-sided dice and taking their sum? For the dice at right, the result would be a 4 (1+3).

Let’s begin with a list of possible sums for two such dice. Since each die has 4 sides, there are 4 x 4 = 16 different pairs.

1 + 1 = 2 2 + 1 = 3 3 + 1 = 4 4 + 1 = 5

1 + 2 = 3 2 + 2 = 4 3 + 2 = 5 4 + 2 = 6

1 + 3 = 4 2 + 3 = 5 3 + 3 = 6 4 + 3 = 7

1 + 4 = 5 2 + 4 = 6 3 + 4 = 7 4 + 4 = 8

More Examples – 2 dice

Now for the probability model. The possible sums then range from 2 to 8. Now count the numbers for each sum and divide them by 16:

1 + 1 = 2 2 + 1 = 3 3 + 1 = 4 4 + 1 = 5

1 + 2 = 3 2 + 2 = 4 3 + 2 = 5 4 + 2 = 6

1 + 3 = 4 2 + 3 = 5 3 + 3 = 6 4 + 3 = 7

1 + 4 = 5 2 + 4 = 6 3 + 4 = 7 4 + 4 = 8

Outcome 2 3 4 5 6 7 8

Probability 1/16 2/16 3/16 4/16 3/16 2/16 1/16

As decimal .063 .125 .188 .250 .188 .125 .063